New Lie systems from Goursat distributions: reductions and reconstructions

TL;DR

This paper explores new classes of Lie systems derived from Goursat distributions, focusing on the n-trailer system, and discusses methods for symmetry reduction and solution reconstruction to extend Lie system applications.

Contribution

It introduces a novel connection between Goursat distributions and Lie systems, enabling symmetry reductions and solution reconstructions for broader classes of differential equations.

Findings

The n-trailer system relates to a Lie system for n=0,1.

Symmetry reduction techniques are applicable to these systems.

Reconstruction methods facilitate solving original systems from reduced ones.

Abstract

We show that types of bracket-generating distributions lead to new classes of Lie systems with compatible geometric structures. Specifically, the $n$-trailer system is analysed, showing that its associated distribution is related to a Lie system if $n = 0$ or $n = 1$. These systems allow symmetry reductions and the reconstruction of solutions of the original system from those of the reduced one. The reconstruction procedure is discussed and indicates potential extensions for studying broader classes of differential equations through Lie systems and new types of superposition rules.